Question

begin by graphing f(x) = \log_{2}x. then use transformations of this graph to graph the given function. what is the vertical asymptote? use the graphs to determine the given functions domain and range.
\quad g(x) = \log_{2}(x + 5)
\\
dashed line. use the graphing tool to graph the equations

what is the vertical asymptote of g(x)?
\quad x = -5
(type an equation.)
what is the domain of g(x) = \log_{2}(x + 5)?
\quad (-5, \infty)
(simplify your answer. type your answer in interval notation.)
what is the range of g(x) = \log_{2}(x + 5)?
(simplify your answer. type your answer in interval notation.)

Explanation:

Step1: Recall Log Function Transformations

For a logarithmic function \( f(x) = \log_b(x) \), the transformation \( g(x) = \log_b(x + h) \) is a horizontal shift. The vertical asymptote of \( \log_b(x) \) is \( x = 0 \). For \( g(x) = \log_2(x + 5) \), we set the argument \( x + 5 = 0 \) to find the vertical asymptote.

Step2: Find Vertical Asymptote

Solve \( x + 5 = 0 \), so \( x = -5 \). This is the vertical asymptote.

Step3: Determine Domain

The domain of a logarithmic function \( \log_b(u) \) requires \( u > 0 \). For \( g(x) = \log_2(x + 5) \), solve \( x + 5 > 0 \), which gives \( x > -5 \), or in interval notation \( (-5, \infty) \).

Step4: Determine Range

Logarithmic functions of the form \( \log_b(x + h) \) (where \( b > 0, b
eq 1 \)) have a range of all real numbers, because as \( x \) approaches the vertical asymptote from the right, the function values go to \( -\infty \), and as \( x \) increases, the function values go to \( \infty \). So the range is \( (-\infty, \infty) \).

Answer:

Vertical Asymptote: \( x = -5 \)
Domain: \( (-5, \infty) \)
Range: \( (-\infty, \infty) \)